Version of 31.8.09 Chapter 49 Further topics I conclude the volume with notes on six almost unconnected special topics. In §491 I look at equidis- tributed sequences and the ideal Z of sets with asymptotic density zero. I give the principal theorems on the existence of equidistributed sequences in abstract topological measure spaces, and examine the way in which an equidistributed sequence can induce an embedding of a measure algebra in the quotient algebra PN/Z. The next three sections are linked. In §492 I present some forms of ‘concentration of measure’ which echo ideas from §476 in combinatorial, rather than geometric, contexts, with theorems of Talagrand and Maurey on product measures and the Haar measure of a permutation group. In §493 I show how the ideas of §§449, 476 and 492 can be put together in the theory of ‘extremely amenable’ topological groups. Some of the important examples of extremely amenable groups are full groups of measure-preserving automorphisms of measure algebras, previously treated in §383; these are the subject of §494, where I look also at some striking algebraic properties of these groups. In §495, I move on to Poisson point processes, with notes on disintegrations and some special cases in which they can be represented by Radon measures. In §496, I revisit the Maharam submeasures of Chapter 39, showing that various ideas from the present volume can be applied in this more general context. In §497, I give a version of Tao’s proof of Szemer´edi’s theorem on arithmetic progressions, based on a deep analysis of relative independence, as introduced in §458. Finally, in §498 I give a pair of simple, but perhaps surprising, results on subsets of sets of positive measure in product spaces. Version of 14.8.08 491 Equidistributed sequences In many of the most important topological probability spaces, starting with Lebesgue measure (491Xg), there are sequences which are equidistributed in the sense that, in the limit, they spend the right proportion of their time in each part of the space (491Yf). I give the basic results on existence of equidistributed sequences in 491E-491H, 491Q and 491R. Investigating such sequences, we are led to some interesting properties of the asymptotic density ideal Z and the quotient algebra Z = PN/Z (491A, 491I-491K, 491P). For ‘effectively regular’ measures (491L-491M), equidistributed sequences lead to embeddings of measure algebras in Z (491N). 491A The asymptotic density ideal (a) If I is a subset of N, its upper asymptotic density is ∗ 1 1 d (I) = limsupn→∞ n (I ∩ n), and its asymptotic density is d(I) = limn→∞ n #(I ∩ n) if this is defined. It is easy to check that d∗ is a submeasure on PN (definition: 392A), so that Z = {I : I ⊆ N, d∗(I) = 0} = {I : I ⊆ N, d(I) = 0} is an ideal, the asymptotic density ideal. (b) Note that Z = {I : I N, lim 2−n#(I ∩ 2n+1 \ 2n) = 0}. ⊆ n→∞ PPP If I ⊆ N and d∗(I) = 0, then 2−n#(I ∩ 2n+1 \ 2n) ≤ 2 · 2−n−1#(I ∩ 2n+1) → 0 Extract from Measure Theory, by D.H.Fremlin, University of Essex, Colchester. This material is copyright. It is issued under the terms of the Design Science License as published in http://dsl.org/copyleft/dsl.txt. This is a de- velopment version and the source files are not permanently archived, but current versions are normally accessible through https://www1.essex.ac.uk/maths/people/fremlin/mt.htm. For further information contact [email protected]. c 2009 D. H. Fremlin c 2002 D. H. Fremlin 1 2 Further topics 491Ab −n n+1 n as n → ∞. In the other direction, if limn→∞ 2 #(I ∩ 2 \ 2 ) = 0, then for any ǫ > 0 there is an k+1 k k m m ∈ N such that #(I ∩ 2 \ 2 ) ≤ 2 ǫ for every k ≥ m. In this case, for n ≥ 2 , take kn such that 2kn ≤ n< 2kn+1, and see that 1 k #(I ∩ n) ≤ 2−kn (#(I ∩ 2m)+ n 2kǫ) ≤ 2−kn #(I ∩ 2m) + 2ǫ → 2ǫ n k=m P as n → ∞, and d∗(I) ≤ 2ǫ; as ǫ is arbitrary, I ∈Z. QQQ (c) Writing D for the domain of d, 1 1 D = {I : I ⊆ N, lim sup #(I ∩ n) = liminf #(I ∩ n)} n→∞ n n→∞ n = {I : I ⊆ N, d∗(I) = 1 − d∗(N \ I)}, N ∈D, if I, J ∈D and I ⊆ J then J \ I ∈D, if I, J ∈D and I ∩ J = ∅ then I ∪ J ∈D and d(I ∪ J)= d(I)+ d(J). It follows that if I⊆D and I ∩ J ∈I for all I, J ∈I, then the subalgebra of PN generated by I is included in D (313Ga). But note that D itself is not a subalgebra of PN (491Xa). (d) The following elementary fact will be useful. If hlnin∈N is a strictly increasing sequence in N such that limn→∞ ln+1/ln = 1, and I ⊆ R, then ∗ 1 d (I) ≤ lim supn→∞ #(I ∩ ln+1 \ ln). ln ln +1− 1 PPP Set γ = limsupn→∞ #(I ∩ ln+1 \ ln), and take ǫ > 0. Let n0 be such that #(I ∩ ln+1 \ ln) ≤ ln ln +1− (γ + ǫ)(ln+1 − ln) and ln+1 − ln ≤ ǫln for every n ≥ n0, and write M for #(I ∩ ln0 ). If m>ln0 , take k such that lk ≤ m<lk+1; then k−1 #(I ∩ m) ≤ M + #(I ∩ ln+1 \ ln)+(m − lk) nX=n0 k−1 ≤ M + (γ + ǫ)(ln+1 − ln)+ lk+1 − lk ≤ M + m(γ + ǫ)+ ǫm, nX=n0 so 1 M #(I ∩ m) ≤ + γ + 2ǫ. m m Accordingly d∗(I) ≤ γ + 2ǫ; as ǫ is arbitrary, d∗(I) ≤ γ. QQQ *(e) The following remark will not be used directly in this section, but is one of the fundamental properties of the ideal Z. If hInin∈N is any sequence in Z, there is an I ∈ Z such that In \ I is finite for every n. PPP Set Jn = j≤n Ij for each n, so that hJnin∈N is a non-decreasing sequence in Z. Let hlnin∈N be a strictly −n increasingS sequence in N such that, for each n, #(Jn ∩ k) ≤ 2 k for every k ≥ ln. Set I = n∈N Jn \ ln. Then In \ I ⊆ ln is finite for each n. Also, if n ∈ N and ln ≤ k<ln+1, S −n #(I ∩ k) ≤ #(Jn ∩ k) ≤ 2 k, so I ∈Z. QQQ 491B Equidistributed sequences Let X be a topological space and µ a probability measure on X. ∗ I say that a sequence hxnin∈N in X is (asymptotically) equidistributed if µF ≥ d ({i : xi ∈ F }) for 1 every measurable closed set F ⊆ X; equivalently, if µG ≤ lim infn→∞ n #({i : i < n, xi ∈ G}) for every measurable open set G ⊆ X. Measure Theory 491C Equidistributed sequences 3 Remark Equidistributed sequences are often called uniformly distributed. Traditionally, such sequences have been defined in terms of their action on continuous functions, as in 491Cf. I have adopted the definition here in order to deal both with Radon measures on spaces which are not completely regular (so that we cannot identify the measure with an integral) and with Baire measures (so that there may be closed sets which are not measurable). Note that we cannot demand that the sets {i : xi ∈ F } should have well-defined densities (491Xi). 491C I work through a list of basic facts. The technical details (if we do not specialize immediately to metrizable or compact spaces) are not quite transparent, so I set them out carefully. Proposition Let X be a topological space, µ a probability measure on X and hxnin∈N a sequence in X. 1 n N (a) hxnin∈ is equidistributed iff fdµ ≤ lim infn→∞ n+1 i=0 f(xi) for every measurable bounded lower semi-continuous function f : X → RR. P 1 n N (b) If µ measures every zero set and hxnin∈ is equidistributed, then limn→∞ n+1 i=0 f(xi) = fdµ for every f ∈ Cb(X). P R 1 n (c) Suppose that µ measures every zero set in X. Iflimn→∞ n+1 i=0 f(xi)= fdµ for every f ∈ Cb(X), ∗ then d ({n : xn ∈ F }) ≤ µF for every zero set F ⊆ X. P R (d) Suppose that X is normal and that µ measures every zero set and is inner regular with respect to the 1 n N closed sets. If limn→∞ n+1 i=0 f(xi)= fdµ for every f ∈ Cb(X), then hxnin∈ is equidistributed. (e) Suppose that µ is τ-additiveP and thereR is a base G for the topology of X, consisting of measurable 1 sets and closed under finite unions, such that µG ≤ lim infn→∞ n+1 #({i : i ≤ n, xi ∈ G}) for every G ∈G. Then hxnin∈N is equidistributed. (f) Suppose that X is completely regular and that µ measures every zero set and is τ-additive. Then 1 n N hxnin∈ is equidistributed iff the limit limn→∞ n+1 i=0 f(xi) is defined and equal to fdµ for every f ∈ Cb(X). P R (g) Suppose that X is metrizable and that µ is a topological measure. Then hxnin∈N is equidistributed 1 n iff the limit limn→∞ n+1 i=0 f(xi) is defined and equal to fdµ for every f ∈ Cb(X). (h) Suppose that X isP compact, Hausdorff and zero-dimensional,R and that µ is a Radon measure on X. Then hxnin∈N is equidistributed iff d({n : xn ∈ G})= µG for every open-and-closed subset G of X.